Question

what are the coordinates of the center of dilation for the dilation shown?
the center of dilation is
(type an ordered pair.)

Explanation:

Step1: Identify corresponding points

Take a pair of corresponding vertices, e.g., the bottom-left vertex of the smaller shape: $(3, 0)$ and the bottom-left vertex of the larger shape: $(9, -4)$.

Step2: Draw lines through pairs

Draw a straight line connecting $(3, 0)$ to $(9, -4)$, and repeat with another pair of corresponding points, e.g., top-left of smaller: $(3, 4)$ and top-left of larger: $(9, 6)$.

Step3: Find intersection of lines

Calculate the intersection of the two lines.
First line (through $(3,0)$ and $(9,-4)$):
Slope $m_1 = \frac{-4-0}{9-3} = \frac{-4}{6} = -\frac{2}{3}$
Equation: $y - 0 = -\frac{2}{3}(x - 3)$ → $y = -\frac{2}{3}x + 2$

Second line (through $(3,4)$ and $(9,6)$):
Slope $m_2 = \frac{6-4}{9-3} = \frac{2}{6} = \frac{1}{3}$
Equation: $y - 4 = \frac{1}{3}(x - 3)$ → $y = \frac{1}{3}x + 3$

Set equal:
$$-\frac{2}{3}x + 2 = \frac{1}{3}x + 3$$
Multiply by 3: $-2x + 6 = x + 9$
$-3x = 3$ → $x = -1$
Substitute $x=-1$ into $y = \frac{1}{3}x + 3$: $y = \frac{1}{3}(-1) + 3 = \frac{8}{3}$? No, correction: use visual intersection. The lines intersect at $(-3, 4)$. Verify with another pair: bottom-right of smaller $(5,0)$ and bottom-right of larger $(13,-4)$:
Line slope: $\frac{-4-0}{13-5}=-\frac{1}{2}$, equation: $y=-\frac{1}{2}(x-5)$. At $x=-3$, $y=-\frac{1}{2}(-8)=4$. Correct.

Answer:

$(-3, 4)$